Problem: You have found the following ages (in years) of all 5 lions at your local zoo: $ 6,\enspace 11,\enspace 3,\enspace 5,\enspace 4$ What is the average age of the lions at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we have data for all 5 lions at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{6 + 11 + 3 + 5 + 4}{{5}} = {5.8\text{ years old}} $ Find the squared deviations from the mean for each lion. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $6$ years $0.2$ years $0.04$ years $^2$ $11$ years $5.2$ years $27.04$ years $^2$ $3$ years $-2.8$ years $7.84$ years $^2$ $5$ years $-0.8$ years $0.64$ years $^2$ $4$ years $-1.8$ years $3.24$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{0.04} + {27.04} + {7.84} + {0.64} + {3.24}} {{5}} $ $ {\sigma^2} = \dfrac{{38.8}}{{5}} = {7.76\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{7.76\text{ years}^2}} = {2.8\text{ years}} $ The average lion at the zoo is 5.8 years old. There is a standard deviation of 2.8 years.